Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $t \neq 0$. $a = \dfrac{2t}{8(4t + 7)} \div \dfrac{-7}{32t + 56} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $a = \dfrac{2t}{8(4t + 7)} \times \dfrac{32t + 56}{-7} $ When multiplying fractions, we multiply the numerators and the denominators. $a = \dfrac{ 2t \times (32t + 56) } { 8(4t + 7) \times -7 } $ $ a = \dfrac {2t \times 8(4t + 7)} {-7 \times 8(4t + 7)} $ $ a = \dfrac{16t(4t + 7)}{-56(4t + 7)} $ We can cancel the $4t + 7$ so long as $4t + 7 \neq 0$ Therefore $t \neq -\dfrac{7}{4}$ $a = \dfrac{16t \cancel{(4t + 7})}{-56 \cancel{(4t + 7)}} = -\dfrac{16t}{56} = -\dfrac{2t}{7} $